Determining the axial direction of high-shear flux transfer events ...

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117, A09220, doi:10.1029/2012JA017831, 2012

Determining the axial direction of high-shear flux transfer events: Implications for models of FTE structure R. C. Fear,1 S. E. Milan,1 and K. Oksavik2 Received 14 April 2012; revised 19 July 2012; accepted 1 August 2012; published 18 September 2012.

[1] Flux transfer events are bursts of dayside reconnection, which give rise to local perturbations of the magnetic field that can be observed by spacecraft near the magnetopause. Although it is commonly accepted that flux transfer events are caused by reconnection, various models exist to explain their structure. A key difference between mechanisms is that when the magnetic shear across the magnetopause is close to 180 , some models will give rise to a structure whose axis is oriented north-south, whereas others will result in a dawn-dusk-oriented structure. Several techniques for determining the axial direction of such structures have been suggested: minimum variance analysis on the magnetic field, minimization of the axial electric field, and Grad-Shafranov reconstruction. We apply these techniques to a series of flux transfer events observed by Cluster at a high-shear magnetopause crossing on 27 March 2007. Minimum variance analysis on the signatures caused by the draping of unreconnected magnetic flux around the flux transfer events consistently results in an axial direction that is dawn-dusk oriented. However, the electric field technique, applied to flux transfer events that are penetrated by the spacecraft, results in a mixture of north-south and dawn-dusk axes. Testing these axial directions with Grad-Shafranov reconstruction suggests that the axes of events which appear to be oriented dawn-dusk might be more reliably determined than the axes of events which appear to be north-south. Using the Grad-Shafranov method alone to determine the axial direction of the penetrated events gives dawn-dusk axial directions, consistent with the results of minimum variance analysis on the draping events. Overall, these observations are consistent with the formation of flux transfer events by processes involving relatively long reconnection lines (e.g., single or multiple X-line reconnection) rather than elbow-shaped flux tubes, as originally envisaged. Furthermore, examination of one particular flux transfer event that was observed by Cluster 1 but not by Cluster 2 (which was closer to the magnetopause) allows us to preclude the elbow-shaped model for this event. Citation: Fear, R. C., S. E. Milan, and K. Oksavik (2012), Determining the axial direction of high-shear flux transfer events: Implications for models of FTE structure, J. Geophys. Res., 117, A09220, doi:10.1029/2012JA017831.

1. Introduction [2] Since the discovery of in situ flux transfer event (FTE) signatures by Russell and Elphic [1978, 1979], several conceptual models have been proposed to explain their characteristic magnetic field and plasma variations. Russell and Elphic [1978] proposed an elbow-shaped flux tube model, where FTE structures were formed by a localized patch of reconnection, resulting in a pair of flux tubes that are broadly oriented north-south when the interplanetary 1

Department of Physics and Astronomy, University of Leicester, Leicester, UK. 2 Department of Physics and Technology, University of Bergen, Bergen, Norway. Corresponding author: R. C. Fear, Department of Physics and Astronomy, University of Leicester, Leicester LE1 7RH, UK. ([email protected]) ©2012. American Geophysical Union. All Rights Reserved. 0148-0227/12/2012JA017831

magnetic field (IMF) is strongly southward. Subsequently, Lee and Fu [1985] proposed that multiple X-lines, which could be significantly longer than in the elbow-shaped flux tube model, could form east-west oriented flux tubes which could give rise to the same signatures. Southwood et al. [1988] and Scholer [1988] suggested that bulges could form in the magnetopause boundary layer as a result of the increased thermal pressure of plasma heated by a burst of reconnection at a single X-line, which again could be either long or short (the “single X-line” model). These bulges do not form flux ropes, but still replicate the observed signatures and have the same axial orientation as multiple X-line flux ropes. (See also Scholer [1995], Lockwood and Hapgood [1998] and Fear et al. [2008] for further explanations of these and other FTE models.) [3] A few authors have examined the differences between these models; Lockwood and Hapgood [1998] found that the continuous variation in the ion distribution observed within one example FTE was consistent with the passage through

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open magnetic field lines corresponding to varying elapsed times since reconnection, which they concluded was consistent with the single X-line model and not the elbow-shaped flux tube model. By considering other plasma signatures observed during this event, they also excluded the multiple X-line mechanism as the cause for this FTE. Observations of the velocities of larger-scale FTEs observed several Earth radii from the subsolar point [Fear et al., 2007] have been more consistent with the velocity expected of open magnetic field lines [Cowley and Owen, 1989; Cooling et al., 2001] than the magnetosheath velocity, suggestive of the single X-line or elbow-shaped flux tube mechanism. However, other studies based on reconstruction techniques [Sonnerup et al., 2004; Hasegawa et al., 2006], modeled/ observed seasonal dependencies [Raeder, 2006; Korotova et al., 2008; Fear et al., 2012], comparison of the modeled/ observed details of FTE signatures [Ku and Sibeck, 1997, 1998a, 1998b, 2000; Fear et al., 2010a] and case studies [Hasegawa et al., 2010; Øieroset et al., 2011] suggest a multiple X-line mechanism. (It might also be that the sheardriven 3-D reconnection generation mechanism described by Dorelli and Bhattacharjee [2009] could explain some of the above observations, such as the observed bias toward FTE signatures being observed in the winter hemisphere; however Dorelli and Bhattacharjee [2009] only considered equinoctial conditions, so no conclusions about seasonal bias can be drawn from their work yet [J. Dorelli, private communication].) Hasegawa et al. [2010] and Øieroset et al. [2011] reported converging reconnection jets either side of an FTE, indicating the presence of nearby X-lines on each side. Both events were magnetosheath FTEs; Hasegawa et al. [2010] also reported a heated magnetosheath electron population both parallel and antiparallel to the magnetic field direction, which is inconsistent with a single X-line geometry (unless some other process causes the mirroring of the heated plasma) given the location of the spacecraft outside the magnetopause. However, bidirectional magnetosheath electron signatures appear to be common features of magnetosheath FTEs [Robert et al., 2006; Fear et al., 2007]. [4] The elbow-shaped flux tube model is necessarily patchy, whereas the single and multiple X-line model structures may either be very localized or may extend over a significant portion of the magnetopause, explaining largescale ionospheric signatures of FTEs such as those observed by Milan et al. [2000] as well as smaller-scale ionospheric signatures [e.g., Marchaudon et al., 2004; Oksavik et al., 2004, 2005]. One way to investigate the in situ extent of FTE signatures is to use data from magnetopause crossings made by the four Cluster spacecraft since 2006, as they have regularly crossed the magnetopause with inter-spacecraft separations of the order of 10,000 km. Such a separation is large enough to observe macroscale differences between the signatures observed at the four spacecraft, but small enough to be confident that all spacecraft are observing the same FTE structure rather than another nearby event. Fear et al. [2008] determined the poleward and minimum azimuthal scale sizes of a series of FTEs; while they found that the FTEs clearly had minimum azimuthal scales that were greater than their poleward scale sizes, the relatively low shear between the magnetosheath and magnetospheric magnetic fields (100 ) meant that these structures could still be consistent with the elbow-shaped model. Therefore,

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it is of interest to examine magnetopause crossings which occur when the magnetic shear is much closer to 180 . Under these circumstances, the orientation of the flux tube/ bulge structure should be largely north-south in the elbowshaped flux tube model, and dawn-dusk in the single and multiple X-line models [Sonnerup, 1988]. Therefore, Fear et al. [2010b] applied the same multispacecraft techniques to FTEs observed on a magnetopause crossing in March 2007, which occurred near noon MLT and during which the lagged IMF was mostly strongly southward, resulting in a high magnetic shear at the magnetopause. The separation of the spacecraft was smaller than in the previous case study (8500 km), reducing the minimum azimuthal scale size that can be inferred from the spacecraft separation. While some signatures were observed at all four spacecraft and had a minimum azimuthal scale size which was comparable to or greater than the poleward size (as far as could be deduced from the slightly smaller tetrahedron), interestingly other signatures were observed at only subsets of the spacecraft. Since all four spacecraft were oriented close to a plane that was tangential to the magnetopause, and all four spacecraft crossed the magnetopause once only and within 4 minutes of each other, this suggests that some of the FTE structures were much patchier. [5] The object of this paper is to take advantage of the high magnetic shear at the magnetopause during the interval studied by Fear et al. [2010b], which implies a 90 difference between the axial direction predicted by the elbowshaped flux tube model on one hand and the single and multiple X-line models on the other, to deduce the axial directions of the FTE structures (particularly for the patchy events). Fear et al. [2010b] showed that the larger-scale structures (observed by all four spacecraft) had velocities which were strongly poleward, indicative of axes that are directed azimuthally. However, there were evident variations between some signatures observed at the four spacecraft, and multispacecraft comparisons do not allow conclusions about the orientation of patchy events. Therefore, in this study we will use single-spacecraft methods to determine the axial direction. Several such methods have been suggested, which will be discussed in the following section. In section 3, we briefly outline the instrumentation used in this study. In section 4, we provide an overview of FTEs observed by Fear et al. [2010b], and apply the single spacecraft methods to these FTEs in section 5. We then compare the results from the different methods, discuss our findings and summarize our conclusions.

2. Single-Spacecraft Methods 2.1. Minimum Variance Analysis [6] Minimum variance analysis (MVA) [Sonnerup and Scheible, 1998, and references therein] is a technique that is applied to a vector time series to find the directions in which the component of the vector varies the least and the most (the minimum and maximum variance directions). The minimum and maximum variance directions are orthogonal to each other, and the direction that is orthogonal to both is referred to as the intermediate variance direction. These directions are found by solving an eigenvalue equation which is given as equation (8.7) in Sonnerup and Scheible [1998]. The eigenvectors ^x 1 , ^x 2 and ^x 3 , corresponding to

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eigenvalues l1, l2 and l3 (where l1 > l2 > l3) give the maximum, intermediate and minimum variance directions. [7] Minimum variance analysis on magnetic field data is commonly used to determine the normal to rotational and tangential discontinuities such as the magnetopause current layer [e.g., Sonnerup and Cahill 1967], but several studies have applied the same mathematical technique to determine the axial direction of flux ropes at the magnetopause, in the magnetotail and in the Venusian ionosphere [e.g., Sibeck et al., 1984; Elphic et al., 1980; Elphic and Russell, 1983; Farrugia et al., 1987; Slavin et al., 1989; Zong et al., 1997]. The eigenvector which corresponds to the axial direction depends upon the model that is assumed for the flux rope structure and on the depth of the crossing (measured by the impact parameter, which is defined as the smallest distance between the spacecraft track and the flux rope axis). Several studies have considered flux ropes with a strong axial core field, and where the spacecraft crossed close to the center of the flux rope (i.e., with a small impact parameter) [e.g., Sibeck et al., 1984; Elphic and Russell, 1983; Zong et al., 1997]. In these studies, the maximum variance direction has been taken to be the axial direction, due to the strength of the axis-aligned core field. Slavin et al. [1989] modeled force-free flux ropes, and found that the axial direction corresponded to the intermediate direction due to the relatively weak axial magnetic field. Farrugia et al. [1987] modeled the draping of magnetic flux around a cylinder, assuming incompressible plasma flow around the cylinder, and found that the component of the draped field that was along the axial direction was constant. Consequently, if MVA is applied to a “draping” FTE signature (where there is no electron or ion signature, indicating that the spacecraft did not cross onto open magnetic field lines, or come within two electron gyroradii of doing so [e.g., Oksavik et al., 2002]), then the minimum variance direction should be parallel or antiparallel to the axial direction. [8] Walthour et al. [1993, 1994] and Khrabrov and Sonnerup [1998] investigated the effect of removing the incompressible flow assumption and found a relatively simple expression for an angle by which the minimum variance eigenvector should be rotated to produce an estimate of the axial direction which accounts for the compressibility of the plasma. They showed that if a spacecraft observed only the draping of flux around an FTE, the unperturbed magnetic field outside the FTE (B0) was approximately equal to the mean magnetic field during the passage of the structure (〈B〉), and the maximum and intermediate eigenvalues were well determined with respect to each other, then the magnetopause normal should be well approximated by either the maximum or the intermediate eigenvector and the true axial direction should be given by a clockwise rotation of the minimum eigenvector by an angle qN about the eigenvector which approximates the normal. If the maximum eigenvector approximates the normal (Case 1), then:

where tan y ¼ ðB0 ⋅^x 3 Þ=ðB0 ⋅^x 1 Þ . In the case where l1 ≈ l2, the maximum and intermediate eigenvectors are poorly determined, but this situation corresponds to qN ≈ 0. [9] More recently, Xiao et al. [2004] have modeled forcefree and strong-core-field flux ropes, and carried out minimum variance analysis on the magnetic field observed by various virtual satellites at a range of impact parameters. If the satellite passes through a force-free flux rope [e.g., Slavin et al., 1989], then Xiao et al. [2004] found that the intermediate variance direction was indeed the best predictor of the axial direction, but the difference between the intermediate direction and the true axial direction was up to 20 . When the satellite passes through a flux rope with a primarily axis-aligned current (i.e., non-force-free) then Xiao et al. [2004] found that the maximum variance direction best predicted the axial direction for crossings with small impact parameters, and the intermediate variance direction best predicted it for crossings with larger impact parameters, but in both cases there was a difference between the MVA direction and the true axial direction of 20–30 . In the case of non-force-free flux ropes, more accurate determinations of the axis were obtained by carrying out MVA on the current structure.

tan qN ¼ tan yðl2 =l1 1Þ;

If the structure is in magnetohydrostatic equilibrium, then the pressure gradient balances the magnetic tension transverse to the magnetic field (rt p = (j B)t). Under the 2-D assumption, this reduces to the Grad-Shafranov equation:

ð1Þ

^ 3 Þ=ðB0 ^ where tan y ¼ ðB0 x x 2 Þ and l1 > l2 ≫ l3. If the intermediate eigenvector approximates the normal (Case 2), then the rotation is given by: tan qN ¼ tan yðl1 =l2 1Þ;

2.2. Grad-Shafranov Method [10] Grad-Shafranov reconstruction is a technique that has been proposed to reconstruct two dimensional magnetic field structures from time series data taken from a satellite pass through the structure [Hau and Sonnerup, 1999; Hu and Sonnerup, 2003; Sonnerup et al., 2004, 2006; Hasegawa et al., 2005, 2006]. The technique makes several assumptions. Firstly, it is assumed that the structure is approximately two dimensional (defining the ^x-^y plane, where ^x is along the direction of the spacecraft’s path), such that spatial changes out of the 2-D plane (referred to as the invariant axis ^z ) are much more gradual than within the plane. Secondly, a frame of reference must exist in which the structure is approximately stationary. (This frame is called the “proper frame”, which is a generalization of the de Hoffmann-Teller [dHT] frame [de Hoffmann and Teller, 1950].) Thirdly, it is assumed that inertia effects can be neglected. If the 2-D criterion is met, then the magnetic field can be expressed in terms of the vector potential A ¼ Aðx; yÞ^z [Hau and Sonnerup, 1999]: B ¼ ð∂A=∂y; ∂A=∂x; Bz ðx; yÞÞ:

ð3Þ

The component of the vector potential along the spacecraft track can be obtained by integrating the component of the magnetic field that is perpendicular to both the selected invariant axis and the spacecraft track: Z x Z t Aðx; 0Þ ¼ By ðx; 0Þdx ¼ jVHT j By ðtÞdt: 0

ð2Þ 3 of 19

ð4Þ

0

∂2 A ∂2 A dPt þ ¼ m0 ¼ m0 jz ; ∂x2 ∂y2 dA

ð5Þ


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where Pt = ( p + B2z /2m0) is the transverse pressure, and the plasma pressure p and the axial magnetic field component Bz are functions of A alone. [11] In this study we will not apply the full reconstruction method; we simply take advantage of the assumption that the plasma pressure and axial magnetic field component should be functions of A(x, y) alone. We vary our choice of trial axis direction, and find the direction which best orders the relationship between the plasma pressure and axial field component and A, as suggested by Hau and Sonnerup [1999]. 2.3. Faraday’s Law Method [12] Sonnerup and Hasegawa [2005] presented a new method to determine the axial direction of a flux rope, based on Faraday’s Law. Their method requires the spacecraft to observe the flux rope itself (not just the draping of flux around it). Like the Grad-Shafranov method, it assumes that the flux rope structure is essentially two dimensional and that the flux rope is in hydrostatic equilibrium. Under these conditions, the theta component of Faraday’s Law, expressed in cylindrical coordinates, gives ∂Ez/∂r = 0, where ^z is again the invariant axis direction. Sonnerup and Hasegawa [2005] then outlined a method to find the rest frame and axial direction which satisfy these conditions, by finding the eigenvectors of the matrix: M0 ¼ MEB MBB MBE þ MEE ;

ð6Þ

where the covariance matrices MEB = 〈dEdB〉, MBE = 〈dBdE〉, MEE = 〈dEdE〉 and MBB = 〈dBdB〉 are constructed from the deviations of individual measurements of the electric and magnetic fields from their means (dE = (E 〈E〉) and dB = (B 〈B〉)), and MBB is the inverse of MBB. [13] The eigenvalues and eigenvectors of M0 are found, and the minimum variance direction is taken to be the pre^ ). The velocity of the dictor of the true axial direction (k structure is given by: ^ U0 ; V0 ¼ k

ð7Þ

^ U0 ¼ MBB ⋅ MBE ⋅ k:

ð8Þ

where:

[14] Sonnerup and Hasegawa [2005] discuss three quantitative indicators of how well the basic assumptions are ^ should be small compared with |U0|, the satisfied: U0 ⋅k minimum eigenvalue of M0 should be small, and the ratio of the intermediate to minimum eigenvalues should not be small. They applied this method to two flux transfer events which they had previously examined using Grad-Shafranov reconstructions, and calculated axial directions which were in excellent agreement with Grad-Shafranov axes. The minimum eigenvalues of M0 were 0.003 and 0.02 (mV/m)2, and the ratios of the intermediate to minimum eigenvalues ^ jU0 j were 0.02 and were 6.9 and 6.5. The ratios U0 ⋅k= 0.03. However, they cautioned that the Faraday’s Law method can also fail to produce such good agreement, and so must be used carefully.

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[15] More recently, Trenchi et al. [2011] have also applied this method to a flux rope observed by the Double Star TC-1 satellite, calculating an axial direction that was within 3 of parallel to the orientation inferred of a nearby pair of reconnection lines, suggestive of one of the longer X-line mechanisms.

3. Instrumentation [16] In this paper, we seek to apply the methods outlined in the previous section to Cluster 1 and 2 observations of FTEs reported by Fear et al. [2010b]. All Cluster data are obtained from the Cluster Active Archive. [17] We make use of data from the Fluxgate Magnetometer (FGM) instruments [Balogh et al., 2001; Gloag et al., 2010] and the Plasma Electron and Current Experiment (PEACE) [Johnstone et al., 1997; Fazakerley et al., 2010] on all four spacecraft to provide an overview of the interval, identify flux transfer event signatures and identify whether the spacecraft observes the FTE core and/or the draping region. For these purposes, we use spin resolution (4 s) measurements of the magnetic field observed by FGM, and spin resolution pitch angle distributions from the PEACE instrument. [18] In order to calculate the FTE axial directions with the MVA method, we use FGM data at the highest resolution available (22.4 Hz). [19] The Faraday’s Law method requires observations of the electric and magnetic fields in order to construct the matrix M0 (equation (6)). We obtain the electric field in two ways. On Cluster 1 it is found by taking the cross product of the observed ion velocity and the magnetic field (E = v B). The ion velocity is obtained from moments calculated onboard the spacecraft by the Hot Ion Analyzer on the Cluster Ion Spectrometry instrument (CIS-HIA) [Rème et al., 2001; Dandouras et al., 2010]. HIA provides onboard moments, and hence a measure of the plasma bulk velocity, for each spacecraft spin (4 s). Consequently we calculate E using spin resolution HIA and FGM measurements. CIS data are unavailable from Cluster 2, but we use the electric field measured by the Electric Field and Wave (EFW) instrument [Gustafsson et al., 2001; Khotyaintsev et al., 2010]. A 2-D measurement of the electric field is made in the spin plane of the spacecraft at 25 Hz cadence. The third component is then calculated by assuming that there is no component of the electric field parallel to the magnetic field (E B = 0). In order to reduce the error on the computed electric field component, it is only calculated when the magnetic field vector is more than 15 out of the spin plane of the spacecraft and the magnitude of the component of the magnetic field out of the spin plane is larger than 2 nT. [20] The Grad Shafranov method requires the establishment of the de Hoffmann-Teller frame and calculation of the plasma pressure and magnetic field vector potential. Since ion measurements are unavailable from Cluster 2, this is applied only to data from Cluster 1. We use the ion velocity, density and parallel & perpendicular temperatures calculated from the HIA onboard moments and FGM data, both at spin resolution. [21] We use supporting IMF data from the OMNI high resolution data set [King and Papitashvili, 2005] which (for

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Figure 1. An overview of the interval. From top: the electron distribution observed by Cluster 2 and the BL, BM and BN components and magnetic field magnitude observed by all four spacecraft. The bottom panel shows OMNI IMF clock angle with an additional lag of 250 s to account for the propagation of ^ plane by the the IMF through the magnetosheath to the magnetopause, and the angle made in the ^l m magnetic field observed by Cluster 2 (from 05:08 UT onward). this interval) is based on observations by the Wind magnetic field instrument and solar wind experiment [Lepping et al., 1995; Ogilvie et al., 1995].

4. Interval Overview 4.1. Cluster Observations [22] On 27 March 2007, the Cluster spacecraft crossed the dayside magnetopause near local noon and in the northern hemisphere, at (9.2, 1.3, 2.8)GSM RE. An overview of the interval is shown in Figure 1. The top panel shows the electron distribution observed by Cluster 2. Initially a hot, but relatively rarified distribution was observed, which indicates that the spacecraft was in the magnetosphere. After 05:05 UT, Cluster 2 observed a cooler, denser electron population which indicates that the spacecraft crossed the magnetopause at 05:05 UT and entered into the magnetosheath. [23] The next seven panels in Figure 1 show the magnetic field that was observed by all four spacecraft, plotted in boundary normal coordinates [Russell and Elphic, 1978]. ^ component is directed normal to the local magnetoThe n ^ and directed pause surface, ^l is within the plane defined by n

^ is directed dawnward. The boundary norpoleward, and m mal coordinate system was obtained independently for each spacecraft. The magnetopause normal direction was determined by carrying out minimum variance analysis on the full resolution magnetic field data as each spacecraft crossed the ^ was calculated magnetopause. The azimuthal unit vector (m) by taking the cross product of the Earth’s dipole moment ^ and normalizing the result, and ^l is the cross product with n ^ and n ^ . The interval used for MVA, the ratio of the of m intermediate to minimum eigenvalues and the minimum variance direction (^ n ) for each spacecraft and the two other unit vectors are recorded in Table 1. The ^l vectors are largely ^ is largely along -YGSE aligned along the ZGSE direction, m ^ is largely along XGSE, consistent with the location of and n the spacecraft northward of the subsolar point. [24] The BL component (Figure 1) was initially strongly positive at all four spacecraft (BL ≈ +70 nT). As Cluster 2 left the magnetosphere at 05:05 UT, it observed a decrease of BL to approximately zero, followed by a further decrease to a strongly negative value (BL ≈ 50 nT). The time at which each spacecraft crossed the magnetopause can be determined from the reversal in the BL component. Cluster 3

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Table 1. Boundary Normal Coordinates Spacecraft Cluster Cluster Cluster Cluster

1 2 3 4

MVA Interval 05:06–05:10 05:03–05:10 05:03–05:10 05:06–05:10

UT UT UT UT

lint/lmin

^ GSE n

2.14 2.63 2.62 4.42

(0.916, 0.149, 0.372) (0.880, 0.128, 0.457) (0.958, 0.014, 0.285) (0.925, 0.072, 0.373)

(green trace) crossed the magnetopause at about the same time as Cluster 2, whereas Clusters 1 and 4 (black and blue traces respectively) crossed slightly later at 05:08 UT. This is consistent with the orientation of the Cluster tetrahedron, which is shown in the boundary normal coordinate system for Cluster 3 in Figure 2. The spacecraft were aligned in a plane, close to tangential to the nominal magnetopause; the ^ direction was maximum separation of the spacecraft in the n only 1,300 km, with Clusters 3 and 2 as the outermost spacecraft. Clusters 1 and 2 both lay poleward of Clusters 3 and 4, and Clusters 1 and 2 were the most azimuthallyseparated spacecraft (i.e., they were furthest separated in the ^ direction). m [25] The reversal in BL (Figure 1) was much more abrupt at Clusters 1 and 4 than at Clusters 2 and 3, indicating that Clusters 1 and 4 crossed the magnetopause much more quickly than Clusters 2 and 3. The latter two spacecraft appear to have crossed through a boundary layer between the magnetosphere-proper and the magnetosheath, in which a heated magnetosheath electron population was observed (top panel of Figure 1 for Cluster 2, not shown for Cluster 3) and there was a rapid oscillation of the BN component. This period has been studied in more detail by Laitinen et al. [2010], who discussed the presence of reconnection jets in the magnetosheath and concluded that during the magnetopause crossing intervals reconnection continued in a quasicontinuous but non-steady manner. [26] Before the magnetopause crossings, the BM components at all four spacecraft were all close to zero. Once the spacecraft had entered the magnetosheath, the BM components remained small compared with BL except between 05:15 and 05:25 UT, when the BM components at all four spacecraft increased somewhat in magnitude. [27] The BN components immediately before and after the magnetopause crossings were all close to zero, indicative of

^l GSE

^ GSE m (0.007, (0.302, (0.123, (0.213,

0.923, 0.894, 0.922, 0.912,

0.386) 0.331) 0.368) 0.351)

(0.401, 0.356, (0.366, 0.429, (0.257, 0.388, (0.314, 0.404,

0.844) 0.826) 0.885) 0.859)

a successful determination of the magnetopause normals at all four spacecraft, except for a series of mostly positivethen-negative perturbations. These mainly bipolar BN variations are the magnetopause signatures of standard polarity flux transfer events. There was a mixture of drapingonly signatures without any electron signature (e.g., those observed before 04:58 UT) and “core-crossing” signatures, which were accompanied by an electron signature (e.g., the signatures observed by Cluster 2 at 04:59 and 05:02 UT and most of the signatures observed in the magnetosheath). [28] The bottom panel of Figure 1 shows the angle in the ^l-m ^ plane made by the magnetic field vector observed by Cluster 2 (aLM) in red, and the IMF clock angle (qCA) in brown. qCA was determined from the high resolution OMNI data [King and Papitashvili, 2005], which is discussed further below. aLM remained close to 180 from the magnetopause crossing until 05:15 UT, aside from a couple of transient deviations associated with the passage of FTEs. Between 05:15 and 05:25 UT, aLM reduced due to the enhancement in the magnitude of BM. 4.2. IMF Observations [29] The IMF from the OMNI data set is compared with the magnetosheath magnetic field in Figure 3. The top four panels of Figure 3 show the GSM components and magnitude of the IMF and magnetosheath magnetic field. (The magnetic field components observed by Cluster 2 are divided by 8 in order to compare more easily with the spacecraft situated in the solar wind). The fifth panel shows the angle aLM from Figure 1 and the IMF clock angle (qCA). The following two panels show the Earthward component of the solar wind velocity and the solar wind dynamic pressure, and the bottom panel shows the auroral electrojet AU and AL indices.

Figure 2. The orientation of the spacecraft relative to Cluster 3 in the boundary normal coordinate system for Cluster 3. 6 of 19


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Figure 3. The lagged solar wind conditions, propagated to the bow shock by OMNI and then lagged by a further 250 s to account for the expected transport time of plasma through the magnetosheath to the magnetopause. From top: the GSM components and magnitude of the IMF and the magnetosheath magnetic field observed by Cluster 2 (divided by 8), the IMF clock angle (qCA) and the angle made by the magnetosheath magnetic field tangential to the magnetopause plane (aLM, which is defined in the same sense as qCA), the antisunward component of the solar wind speed, the solar wind dynamic pressure and the auroral electrojet AU and AL indices. [30] The OMNI data set consists of IMF observations from various spacecraft which have been propagated to the nose of the bow shock as described by King and Papitashvili [2005]. During this interval, the source of the OMNI IMF values was the Wind spacecraft, which was situated close to the SunEarth line at (198, 10, 16)GSM RE. In order to account for the propagation of plasma from the bow shock to the magnetopause, we apply an additional lag t sh to the OMNI data which we determine from Khan and Cowley [1999, equation (3)]: t sh ¼

ðRBS RMP Þ VBS : ln ðVBS VMP Þ VMP

ð9Þ

[31] RBS & RMP are the subsolar positions of the bow shock and magnetopause and VBS & VMP are the magnetosheath flow speeds just downstream of the bow shock and immediately upstream of the magnetopause. Following Khan and Cowley [1999], we assume a value of VMP (the inflow speed at a reconnection site at the magnetopause) of

20 km s1, and express VBS in terms of the upstream solar wind speed (VBS = 0.28VSW). Since VSW ≈ 470 km s1 during this interval (Figure 3), VBS = 130 km s1. We use the same value of RBS as that used in the propagation of the OMNI data to the bow shock, which is determined by King and Papitashvili [2005] using the Farris and Russell [1994] model (11.5 RE). Finally, we determine the magnetopause stand-off distance from the Shue et al. [1998] model using the observed solar wind conditions (BZ = 5 nT and p = 3 nPa) which gives RMP = 9.2 RE. Substituting these values into equation (9) gives t sh = 250 s. [32] The IMF was initially northward, and turned abruptly southward at 04:40 UT. The IMF then remained strongly southward (indicated by a clock angle that was very close to 180 ) until 05:14 UT. Between 05:14 and 05:25 UT, the IMF BY component increased, causing a decrease in qCA, before returning to a strongly southward orientation. This rotation is very well matched by the enhancements in BY and decrease in aLM at Cluster. (We note that the leading edge of the IMF BY enhancement is observed about a minute earlier than at Cluster. However, this corresponds to the observation

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of an FTE at Cluster, and so the magnetic field measured in the magnetosheath at this instance does not reflect the unperturbed magnetosheath magnetic field.) [33] The auroral electrojet AU and AL indices are known to increase in magnitude following a sharp southward turning in the IMF. Gjerloev et al. [2010] carried out a superposed epoch analysis of AU and AL during intervals when the IMF made an abrupt southward turn, and found that AU and AL began to increase in magnitude 7 minutes and 24 minutes after the IMF change respectively. In the bottom panel of Figure 3 it can be seen that the magnitudes of the AU and AL indices were initially small, but started to increase at about 04:50 UT and 05:00 UT respectively, about 10 and 20 minutes after the southward IMF turn appeared in the Wind data; the delays between the southward turning of the IMF observed in the lagged OMNI data and the responses of the auroral electrojet indices are consistent with the results of Gjerloev et al. [2010], giving confidence in the timing of the southward turning of the IMF observed by OMNI. [34] During this interval, Geotail was situated much closer to Earth, just upstream of the bow shock near the dawn terminator at (1.0, 21, 16)GSM RE, and observed a similar antisunward component of the solar wind velocity (470 km s1). If a lag of 110 s is applied to the Geotail magnetic field observations (to represent the data being propagated back to the bow shock and then lagged by t sh), the magnetosheath BY increase observed by Cluster is well matched by a similar increase at Geotail, but the southward turning of the IMF occurred at 04:57 UT (not shown)—much later than in the OMNI IMF data and after the start of the increase in AU. Since the AU increase is likely to be triggered by the change in IMF, we conclude that the OMNI observations provide a more reliable estimate of the magnetosheath magnetic field direction while Cluster was in the magnetosphere. At the same time, ACE was close to the L1 Lagrangian point at (240, 37, 8)GSM RE and observed a similar solar wind velocity. If the IMF observed by ACE is lagged by 56 minutes (52 minutes to the bow shock and 250 s through the magnetosheath), then the correspondence between the lagged IMF and the magnetosheath magnetic field observed by Cluster is considerably poorer. Therefore, in Figures 1 and 3, we use the OMNI data, lagged as described above. 4.3. Location of Reconnection Site [35] The strongly southward IMF that is observed between 04:40 and 05:14 UT indicates that the magnetosheath magnetic field was highly likely to be strongly southward during this period. This is verified by direct measurements of the magnetosheath magnetic field after 05:05 UT. Consequently, between 04:40 UT and 05:14 UT, the magnetic shear at the magnetopause was very close to 180 . Between 05:14 and 05:25 UT, the magnetosheath magnetic field was southward but with a duskward component. [36] The FTE signatures observed between 04:54 and 05:25 UT were standard polarity events, indicating that they were moving northward [Cowley, 1982] and so the reconnection site was southward of the spacecraft throughout the interval. This is consistent with the expectation for a component reconnection site passing near the subsolar point

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[Gonzalez and Mozer, 1974]. The expected orientation of the X-line can be calculated from Gonzalez and Mozer [1974, equation (1)]: sin b ¼

B2 B1 cos a B21

þ B22 2B1 B2 cos a

12 ;

ð10Þ

where B1 and B2 are the magnitudes of the magnetosheath and magnetospheric magnetic fields respectively, a is the angle between the magnetosheath and magnetospheric magnetic field vectors (equivalent to aLM or qCA in Figure 1) and b is the angle between the magnetospheric magnetic field (or ^l ) and the reconnection line. During the strongly southward IMF period (04:40 to 05:14 UT), a = 180 and so b = 90 ; therefore we would expect a component X-line to ^ and southward be aligned parallel to the GSM Y axis (or m) of the spacecraft. The lowest value of qCA observed during the BY enhancement is 120 ; taking this to be the lowest value of a, and taking B1 = 50 nT and B2 = 70 nT (from Figure 1), gives a minimum value of b = 65 between 05:14 and 05:25 UT. Therefore the largest expected angle between ^ is 25 . During this time, the X-line is the X-line and m expected to be aligned from the [YGSM > 0, ZGSM > 0] quadrant to the [YGSM < 0, ZGSM < 0] quadrant (or equivalently from [L > 0, M < 0] to [L < 0, M > 0]), but still southward of the spacecraft.

5. Results [37] Now that an overview of the interval has been provided, we turn our attention to the application of the singlespacecraft techniques described in section 2 to the FTEs observed on 27 March. As the most noticeable differences in Figure 1 occur between the observations at Clusters 1 and 2, and these two spacecraft are the most azimuthally separated, we will apply the single-spacecraft techniques to the data from these two spacecraft. Since the the maximum separation of the spacecraft in the direction normal to the magnetopause is 1,300 km, which is much less than estimates of the typical scale size of FTE structures normal to the magnetopause [Saunders et al., 1984; Rijnbeek et al., 1984], we expect much of the difference between observations at different spacecraft to be due to differences in FTE structure along the magnetopause rather than due to being observed at different distances from the magnetopause. [38] The electron distribution parallel and antiparallel to the magnetic field and the full-resolution magnetic field observed by Clusters 1 and 2 are shown in Figures 4 and 5, respectively, which highlight the FTEs selected for analysis in this section. The higher flux of electrons at energies greater than 1 keV moving antiparallel to the magnetic field while the spacecraft are in the magnetosheath (e.g., between 05:10 and 05:15 UT in Figure 4) indicates that the spacecraft are northward of the reconnection site, consistent with the presence of standard polarity FTE signatures. FTEs where no electron signature is observed either parallel or antiparallel to the magnetic field are “draping-only” signatures, and are marked by red boxes and numbered D1–D4 on Cluster 1 and D1–D5 on Cluster 2. FTEs which did occur when an electron signature was observed are shown in purple boxes and numbered C1–C9 and C1–C7. Since we are applying

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Figure 4. The FTEs observed by Cluster 1 and selected for analysis in section 5. This figure shows the electron distribution parallel and antiparallel to the magnetic field, and the full resolution magnetic field in boundary normal coordinates. The “draping-only” FTEs are highlighted in red boxes and numbered D1–D4; the “core-crossing” FTEs are denoted by purple boxes and numbered C1–C9. single-spacecraft techniques, this numbering system is not meant to imply a correspondence between signatures at two different spacecraft; for example, signature C9 at Cluster 1 is highly likely to be caused by the passage of the same FTE as signature C7 at Cluster 2. 5.1. Minimum Variance Analysis [39] We start by applying minimum variance analysis to the simplest case—“draping-only” signatures. We identified four draping-only signatures observed by Cluster 1 and five observed by Cluster 2, which are highlighted by red boxes in Figures 4 and 5. [40] The results of the Khrabrov and Sonnerup [1998] analysis for the draping signatures observed by Clusters 1 and 2 are listed in Tables 2 and 3, respectively. Each table lists, from left, (1) the reference number of the FTE; (2) the start and (3) end time used for the minimum variance analysis (also indicated by the left- and right-hand edges of each red box in Figures 4 and 5); (4) the ratios of the intermediate to minimum and maximum to intermediate eigenvalues (l2/l1 and l1/l2, top and bottom rows within each major row

respectively); (5) the minimum, intermediate and maximum eigenvectors (^x 3, ^x 2 and ^x 1—top, middle and bottom rows of each major row respectively) and (6) the angle between each ^ detereigenvector and the magnetopause normal vector n mined for that spacecraft. In each case, ^x 3 is well-determined, with a ratio l2/l3 > 3 in all cases, and only two events having l2/l3 < 10. ^x 1 and ^x 2 are less well determined (aside from being well constrained to lie within the plane perpendicular to ^x 3). The ratio l1/l2 varies between 1.5 and 2.4 at Cluster 1, and between 1.1 and 1.3 at Cluster 2. In each case, ^ x 3 is within 20 of the magnetopause plane. In some cases, one of the eigenvectors is a reasonable approximation of the mag^ are netopause normal (e.g., out of the Cluster 2 events, ^ x2 & n ^ are only 8 apart for D2). only 6 apart for D4 and ^x 1 & n However, in other cases, both eigenvectors fail to pick out the magnetopause normal. This is not surprising, since l1 ≈ l2 in most cases. [41] The tables continue with: (7) the rotation angle qn ^ ) and Case 1 (^x 1 ≈ n ^ ) from calculated for Case 2 (^x 2 ≈ n equations (2) and (1); (8) the unrotated axial direction (^ x 3 ),

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Figure 5. The FTEs observed by Cluster 2 and selected for analysis in section 5. This figure follows the same format as Figure 4. Note that the FTE numbering at Cluster 2 does not directly map to the numbering at Cluster 1. which assumes incompressible flow [Farrugia et al., 1987] and the axes ^z 2 & ^z 1 for compressible flow (Case 2 and Case 1) determined by rotating ^ x 3 clockwise about the x 1 by qn for the relevant case; (9) the angles eigenvector ^ x 2 or ^

between the azimuthal axis in the boundary normal frame ^ ) and the three possible axial derived for the spacecraft (m directions ^x 3 , ^z 2 & ^z 1 and (10) a comparison between the unperturbed magnetic field B0 and the mean magnetic field

Table 2. Results of Khrabrov and Sonnerup [1998] Analysis on Draped FTE Signatures D1–D4 Observed by Cluster 1a Min eigenvector ^x 3 Int eigenvector ^x 2 Max eigenvector ^x 1

Angle ^ to n

qN (Case 2) (Case 1)

MVA Axis ^x 3 Case 2 Axis ^z 2 Case 1 Axis ^z 1

Angle ^ to m

|B0| |〈B〉| Difference

2.4 5.7

(0.010, 0.970, 0.242) (0.987, 0.048, 0.153) (0.160, 0.238, 0.958) (0.028, 0.942, 0.335)

76 99 17 110

3 0.3 -

(0.010, 0.970, 0.242) (0.001, 0.955, 0.295) (0.015, 0.970, 0.243) (0.028, 0.942, 0.335)

14 17 14 20

74.3 nT 71.8 nT 7 74.3 nT

1.5 13.3

(0.639, 0.241, 0.730) (0.768, 0.234, 0.596) (0.158, 0.986, 0.046)

43 53 93

0.5 0.4 -

(0.035, 0.940, 0.340) (0.024, 0.940, 0.340) (0.158, 0.986, 0.046)

20 20 9

72.6 nT 5 74.3 nT

1.8 66.0

(0.796, 0.155, 0.585) (0.584, 0.055, 0.810) (0.306, 0.922, 0.239)

54 36 104

14 5 -

(0.291, 0.945, 0.146) (0.089, 0.996, 0.004) (0.306, 0.922, 0.239)

19 5 23

73.5 nT 4 74.3 nT

1.8

(0.784, 0.386, 0.486) (0.540, 0.039, 0.841)

119 33

22 10

(0.489, 0.867, 0.099) (0.165, 0.975, 0.151)

30 13

75.2 nT 5

l2/l3 FTE

MVA Start (UT)

MVA Stop (UT)

D1

04:50:00

04:53:00

l1/l2 3.4

D2

D3

D4

04:54:00

04:55:15

04:56:30

04:54:15

04:56:30

04:57:30

a

All vectors are provided in the boundary normal coordinate system for Cluster 1.

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Table 3. Results of Khrabrov and Sonnerup [1998] Analysis on Draped FTE Signatures D1–D5 Observed by Cluster 2a l2/l3 FTE

MVA Start (UT)

MVA Stop (UT)

D1

04:52:30

04:54:30

l1/l2 10.4 1.1 38.6

D2

04:54:30

04:55:30 1.3 17.9

D3

04:56:00

04:57:45 1.3 21.8

D4

04:59:45

05:00:20 1.1 16.7

D5

05:02:45

05:03:30 1.1

Min eigenvector ^x 3 Int eigenvector ^x 2 Max eigenvector ^x 1

Angle ^ to n

qN (Case 2) (Case 1)

MVA Axis ^x 3 Case 2 Axis ^z 2 Case 1 Axis ^z 1

Angle ^ to m

|B0| |〈B〉| Difference

(0.464, 0.857, 0.224) (0.656, 0.502, 0.563) (0.595, 0.114, 0.796)

77 125 37

3.7 2.8

(0.464, 0.857, 0.224) (0.501, 0.848, 0.172) (0.432, 0.880, 0.197)

31 32 28

72.6 nT 72.4 nT 0.4

(0.277, 0.958, 0.075) (0.952, 0.284, 0.117) (0.133, 0.039, 0.990)

86 97 8

47 4

(0.277, 0.958, 0.075) (0.286, 0.682, 0.673) (0.210, 0.976, 0.067)

17 47 13

72.6 nT 71.6 nT 0.8

(0.128, 0.982, 0.140) (0.750, 0.188, 0.634) (0.648, 0.024, 0.761)

98 51 41

3 2

(0.128, 0.982, 0.140) (0.162, 0.982, 0.099) (0.099, 0.988, 0.115)

11 11 9

72.6 nT 73.1 nT 0.4

(0.124, 0.992, 0.025) (0.088, 0.036, 0.995) (0.988, 0.121, 0.092)

91 6 85

0.9 19

(0.124, 0.992, 0.025) (0.139, 0.990, 0.024) (0.089, 0.950, 0.298)

7 8 18

72.6 nT 73.6 nT 1.2

(0.101, 0.995, 0.011) (0.818, 0.077, 0.570) (0.566, 0.067, 0.821)

91 125 35

0.5 0.3

(0.101, 0.995, 0.011) (0.106, 0.994, 0.019) (0.096, 0.995, 0.014)

6 6 6

72.6 nT 72.4 nT 1.3

a


within the MVA interval 〈B〉. These two vectors are very similar in all cases (the maximum angular difference being 7 at Cluster 1 and 1.3 at Cluster 2, and the maximum magnitude difference being 3.5%), indicating that the assumptions made by Khrabrov and Sonnerup [1998] are valid for these FTEs. [42] The best estimate for each FTE axis is highlighted in bold in Tables 2 and 3. This is determined as follows: x 1 ⋅^ n > 0:8, then we take the maximum variance [43] 1. If ^ eigenvector to be a reasonable approximation of the magnetopause normal, and the axis ^z 1 is selected.

n > 0:8 , then we take the intermediate [44] 2. If ^x 2 ⋅^ eigenvector to be a reasonable approximation of the normal, and ^z 2 is selected. [45] 3. If neither condition is satisfied, then we take the unrotated minimum variance eigenvector to be the best approximation of the FTE axial direction. [46] These axes are plotted in Figure 6 (Cluster 1) and ^ Figure 7 (Cluster 2); the axes are closely aligned to the m axis with the exception of FTE D1 at Cluster 2. In the cases where neither ^z 1 nor ^z 2 could be selected, the rotation angle qn would have been small (less than 4 ) for either Case 1 or Case 2. This reflects the conclusion of [Walthour et al., 1993, 1994] that maximum and intermediate eigenvectors

Figure 6. The axes determined by minimum variance analysis, including the correction proposed by Khrabrov and Sonnerup [1998], for the “draping-only” FTE signatures D1–D4 observed by Cluster 1.

Figure 7. The axes determined by minimum variance analysis, including the correction proposed by Khrabrov and Sonnerup [1998], for the “draping-only” FTE signatures D1–D5 observed by Cluster 2.

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Table 4. Results of Sonnerup and Hasegawa [2005] Analysis on “Core-Crossing” FTE Signatures C1–C9 Observed by Cluster 1a FTE

MVA Start (UT)

MVA Stop (UT)

l2/l3

^ Min eigenvector k

^ /|U0| U0 k

V0 (km s1)

No. pts.

|rk|

C1 C2 C3 C4 C5 C6 C7 C8 C9

05:02:37 05:04:47 05:07:11 05:08:10 05:08:57 05:09:49 05:14:02 05:16:15 05:22:45

05:02:55 05:05:11 05:07:53 05:08:53 05:09:43 05:10:52 05:14:29 05:16:32 05:23:59

6.6 2.0 4.0 4.0 19.3 2.8 1.1 17.3 2.8

(0.584, 0.701, 0.408) (0.997, 0.048, 0.058) (0.985, 0.149, 0.081) (0.906, 0.213, 0.365) (0.981, 0.191, 0.012) (0.009, 0.997, 0.074) (0.620, 0.772, 0.139) (0.431, 0.613, 0.663) (0.593, 0.805, 0.018)

0.3 0.8 0.08 0.1 0.002 0.2 0.04 0.4 0.01

(420, 96, 434) (10, 194, 16) (12, 71, 12) (24, 4, 62) (51, 28, 2) (159, 0, 12) (117, 94, 2) (229, 65, 89) (85, 62, 18)

5 5 11 10 11 16 7 5 18

0.90 1.00 0.38 0.58 0.26 0.86 0.68 0.70 0.92

a


that are poorly defined with respect to each other correspond to the case when qn is small. [47] We also attempted minimum variance analysis on two of the “core-crossing” FTEs (Cluster 2 FTEs C1 and C2), but obtained results that were inconsistent with the assumptions made in the modeling studies, such as that by Xiao et al. [2004]. Neither FTE exhibited a strong core magnetic field (Figure 5), and so from the results of Xiao et al. [2004] we would predict that the intermediate variance direction should identify the axial direction whether or not a strong core field were present deeper inside the FTE (and hence unobserved). The intermediate variance direction was not as well-defined as we would wish with respect to either of the other two eigenvectors (l2/l3 = 2.5 & 2.8; l1/l2 = 4.3 & 6.2), but the intermediate eigenvector had its strongest ^ direction. If an FTE did have an axial component in the n direction normal to the magnetopause (e.g., at the point where the flux crosses the magnetopause in the Russell and Elphic [1978] model) then the expected BN signature would be monopolar, rather than the bipolar signature that is identified. The failure of MVA to find self-consistent axes in these cases is consistent with the conclusion by Xiao et al. [2004] that minimum variance analysis on the magnetic field is not necessarily the best means of identifying the axis of FTEs if the spacecraft does sample the flux rope itself, rather than just the draping region. 5.2. Faraday’s Law [48] In this section, we apply the method based on Faraday’s Law that was proposed by Sonnerup and Hasegawa [2005]. This method requires that the spacecraft enters onto the flux rope itself, rather than observing only the draping region, and so we apply the method to the “core-crossing” FTEs C1–C9 on Cluster 1 and C1–C7 on Cluster 2. The start and end times used for this method (corresponding to the vertical edges of the purple boxes in Figures 4 and 5) were

selected based on the electron distributions to ensure that the spacecraft was on open field lines throughout. [49] The method involves finding the eigenvectors for the matrix given in equation (6), which depend upon the electric and magnetic fields observed during the FTE. We take a different approach for each of the two spacecraft, in order to compare the relative merits of using different instruments to determine the electric field. For Cluster 1, we calculate the electric field vector from the cross product of the ion velocity and the magnetic field, both of which make direct 3-D measurements of the parameters they observe. However, as a whole spin of the spacecraft is required to make a complete measurement of the ion distribution (from which the velocity is determined), this restricts the analysis to spin cadence data. For Cluster 2, we use the electric field measured by the EFW experiment. This allows use of higher cadence data (the electric field is measured at 25 Hz, and we use the full resolution magnetic field), but the disadvantage is that the out-of-plane component of the electric field is inferred from the assumption E B = 0, and is only available when the conditions detailed in section 3 are satisfied. [50] The results are outlined in Tables 4 and 5. The columns of each table detail: (1) the FTE number; (2) the start and (3) stop time used for the minimum variance analysis; (4) the intermediate to minimum eigenvalue ratio of the ^ , which is the matrix M0; (5) the minimum eigenvector k ^ jU0 j, estimator of the axial direction; (6) the quantity U0 ⋅k= where the velocity U0 is given by equation (8); (7) the proper velocity of the structure V0; and (8) the number of data points in the time series used for MVA. Table 4 also lists (9) a parameter rk that will be discussed in section 5.3. [51] Unlike the results of the minimum variance analysis on the draping signatures, most of the axial directions are directed poleward (shown in Figures 8 and 9), although there are some exceptions. The two main variables that Sonnerup and Hasegawa [2005] proposed to indicate the accuracy of

Table 5. Results of Sonnerup and Hasegawa [2005] Analysis on “Core-Crossing” FTE Signatures C1–C7 Observed by Cluster 2a FTE

MVA Start (UT)

MVA Stop (UT)

l2/l3

^ Min eigenvector k

^ /|U0| U0 ⋅ k

V0 (km s1)

No. pts.

C1 C2 C3 C4 C5 C6 C7

04:59:00 05:01:55 05:03:38 05:08:20 05:10:36 05:13:12 05:23:06

04:59:20 05:02:11 05:03:57 05:08:36 05:11:34 05:13:37 05:23:54

4.6 5.2 25.5 16.2 4.9 2.6 22.6

(0.984, 0.178, 0.025) (0.989, 0.146, 0.025) (0.998, 0.022, 0.054) (0.949, 0.316, 0.014) (0.981, 0.155, 0.117) (0.771, 0.331, 0.544) (0.350, 0.928, 0.126)

0.2 0.02 0.5 0.4 0.3 0.2 0.1

(5, 29, 3) (8, 50, 17) (3, 110, 3) (6, 18, 5) (7, 22, 27) (123, 88, 121) (134, 46, 30)

444 359 426 358 1296 560 1076

a


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Figure 8. The axes determined by the Faraday’s Law method, for the “core-crossing” FTE signatures C1–C9 observed by Cluster 1. the determined axial direction were the eigenvalue ratio ^ jU0 j (since the velocity U0 l2/l3 and the value of U0 ⋅k= should be perpendicular to the axial direction). Only two signatures (C5 on Cluster 1 and C2 on Cluster 2) satisfy the ^ j=jU0 < 0.05 and l2/l3 > 5. Both are criteria that U0 ⋅k events for which the Faraday’s Law method estimates an axial direction that is largely aligned with ^l. [52] If the axes of the “core-crossing” FTEs observed by Cluster 1 are calculated using the Faraday’s Law method with high resolution EFW data from Cluster 1 (not shown), then the results differ from those based on the CIS and FGM data, and are more like the results from Cluster 2: FTE C9 has a similar axis to that obtained using the electric field from CIS and FGM (Figure 8) (also similar to the axis determined from signature C7 at Cluster 2, which is likely to be caused by the passage of the same flux rope), but the other axes are all directed largely along ^l. 5.3. Grad-Shafranov Method [53] Another means of checking the quality of the results in section 5.2 is to apply the first stages of GradShafranov reconstruction. If the axial directions determined in section 5.2 are good, then the total pressure transverse to the axial direction (PT = ( pplasma + B2z /2m0)) should be well ordered as a function of the magnitude of the magnetic field vector potential (A). The function need not be linear; we therefore calculated the Spearman’s Rank correlation coefficient (rk) between these two parameters, which is provided in the final column of Table 4. (Since this requires knowledge of the plasma pressure, it has only been calculated for Cluster 1.) With the exception of the shorter-duration events (with ^ is mainly fewer than ten data points), the events where k poleward give rise to relatively weak correlation coefficients ^ is mostly azimuthal (all less than 0.6) and the events where k

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all give rise to stronger coefficients. This suggests that the poleward axes might not be very well determined, notwithstanding the fact that other two quality parameters may be satisfied (e.g., signature C5). [54] The final method we used to determine the FTE axis was that proposed by Hau and Sonnerup [1999]. First, the proper frame of the structure was calculated using de Hoffmann-Teller analysis. A trial axis was then defined by an ^ plane. The tangential total pressure and angle q within the ^l-m vector potential were calculated and the Spearman’s Rank correlation (r) between these two parameters was calculated. (Again, since this requires knowledge of the plasma pressure, this method was only applied to Cluster 1.) The axis angle q was varied in 1 intervals between 90 , and the angle qMAX which maximized the correlation coefficient (rMAX) was selected as the axial direction. [55] The results of this analysis are provided in Table 6, which lists (1–3) the interval used for the analysis (which is the same as used in section 5.2); (4) the de Hoffmann-Teller velocity in boundary normal coordinates (VHT), which is the velocity of the proper frame used in the analysis; (5) the correlation (rHT) between the observed electric field v B and the electric field of the proper frame VHT B, which indicates the quality of the de Hoffmann-Teller frame; (6) rMAX; (7) qMAX; (8) the axis defined by qMAX in boundary normal coordinates, which is taken to be the axial direction of the FTE and (9) the number of data points used in the analysis. The plots of PT against A for the optimized axial direction for each FTE are shown in Figure 10. In some cases (particularly those with the fewest data points), the correlation coefficient was equal to rMAX over a range of angles. In these cases, the value of qMAX is obtained by calculating the mean axial direction of the values of q which give rise to a correlation of rMAX, as described by Mardia and Jupp [2000, pp. 5, 15].

Figure 9. The axes determined by the Faraday’s Law method, for the “core-crossing” FTE signatures C1–C7 observed by Cluster 2.

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Table 6. Results of the Grad-Shafranov Method on “Core-Crossing” FTE Signatures C1–C9 Observed by Cluster 1a FTE C1 C2 C3 C4 C5 C6 C7 C8 C9

Start (UT) 05:02:37 05:04:47 05:07:11 05:08:10 05:08:57 05:09:49 05:14:02 05:16:15 05:22:45

Stop (UT) 05:02:55 05:05:11 05:07:53 05:08:53 05:09:43 05:10:52 05:14:29 05:16:32 05:23:59

VHT (km s1)

rHT

(95.0, 30.7, 3.5) (243.6, 55.5, 1.7) (359.4, 47.9, 23.2) (215.4, 54.6, 37.6) (154.1, 28.5, 4.4) (183.9, 5.4, 1.1) (183.2, 10.8, 2.5) (206.2, 24.2, 4.6) (198.6, 80.0, 3.5)

0.935 0.812 0.947 0.954 0.946 0.966 0.984 0.996 0.967

rMAX 1.00 1.00 0.97 0.66 0.82 0.89 0.68 1.00 0.97

qMAX

11 4 80 12 66 77 46 23 87

Axis

No. pts.

(0.980, 0.191, 0.000) (0.999, 0.070, 0.000) (0.174, 0.985, 0.000) (0.978, 0.208, 0.000) (0.407, 0.914, 0.000) (0.225, 0.974, 0.000) (0.695, 0.719, 0.000) (0.921, 0.391, 0.000) (0.052, 0.999, 0.000)

5 5 11 10 11 16 7 5 18

a All vectors are given in boundary normal coordinates. In cases where more than one value of q results in a correlation coefficient equal to rMAX, qMAX is the directional mean of all such values of q.

[56] This method also produces a mixture of axial directions (plotted in Figure 11), with some being poleward and others more azimuthal. However, the events with polewarddirected axes (C1, C2, C4, C7 and C8) are either determined from relatively few data points (